Make object follow a path from a specific frame (not frame 0)
Edit: Ok, actually you should rather unparent and give your object a Follow Path constraint with the curve as Target.

Then, when you press Animate Path, it moves your object.

You can ether play with the constraint Offset value to move it the way you want. In your case keep it at 0 at Frame 50.



Or you can select the curve, go in the Graph Editor, unfold the Bezier Curve track on the left, select the Evaluation Time, display the N panel, select the Modifiers tab, close the Generator.



Now keep the curve selected, go in the Properties editor > Data > Path Animation, play with the Evaluation Time value and create some keyframes to make the object move along the curve. In your case, keep the Evaluation Time at 0 until frame 50.